Complement and union of events

From Kolmogorov’s three axioms the most widely used results in practice follow immediately.

Property — First results

For any event $A$ and any pair of events $A, B$:

• $P(\emptyset) = 0$, $\; P(\Omega) = 1$;
• Opposite (complementary) event: $P(\overline{A}) = 1 - P(A)$;
• Union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (subtracting the intersection counted twice).

The union formula subtracts $P(A \cap B)$ because, on adding $P(A)$ and $P(B)$, the outcomes common to both would otherwise be counted twice.

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Topics: Probability
Concepts: Complementary event · Probability · Union of events
Methods: Complementary prob
Skills: Using formulae